| 摘要(英) |
The best choices and results of many games can obtain through intelligent computation. The blackjack game is a simple and most common one. Blackjack games often have different rules in different regions. How the player can execute the best strategy and obtain its corresponding expectation value have been discussed by many books and articles it in the past seventy years. As the expectation value is greater than zero, the player will win the game in a long term, and vice versa. What a player really concerns is whether the expectation value is positive if he already executes the best strategy?
There are two approaches to obtain the best strategy and expectation values. The first approach is to perform computer simulations, and the second approach is to build mathematical models. It is obvious that there are too many permutations in the game. However, under each game rule, the number of simulations must be very large, and the simulation time is too long to be accepted. Moreover, academically we wish the best strategy and the exact analytical expectation value can be obtained. Many articles built mathematical and statistical models to obtain approximate analytical value based on some estimations that will introduce some bias in the result.
This thesis uses the sequence of the poker cards as the input signal. Assume that the cards are shuffled cleanly before the game so that the input signal can be treated as a random signal. Generally speaking, the game uses many decks and the cards will be shuffled after just playing few games to prevent that player can guess the probability of subsequent cards based on the previous appeared cards. Consequently, it can be assumed that the number of poker decks is infinite. This thesis will propose a random signal processing algorithm to determine the best strategy and its corresponding expectation value for the player. First, I apply a tree-based structure algorithm to exhaustive list all the permutations of the dealer’s card, calculate the probability distribution of dealer’s final points and use the recursive algorithm, the best strategy and expectation values without error or deviation under various game rules can be obtained within a second. Next, I will answer whether the expectation value is positive under each game rule.
Finally, the accuracy of the proposed analytical result is confirmed through computer simulations. I found that the expectation values based on these two approaches are asymptotically identical as the number of simulations approaches several billion that shows the correctness of the analytical values. |
| 參考文獻 |
[1] R. R. Baldwin, W. E. Cantey, H. Maisel, and J. P. McDermott, "The optimum strategy in blackjack," Journal of the American Statistical Association, vol. 51, no. 275, pp. 429-439, 1956.
[2] E. Thorp, "A favorable strategy for twenty-one," Proceedings of the National Academy of Sciences of the United States of America, vol. 47, no. 1, p. 110, 1961.
[3] A. Manson, A. Barr, and J. Goodnight, "Optimum zero-memory strategy and exact probabilities for 4-deck blackjack," The American Statistician, vol. 29, no. 2, pp. 84-88, 1975.
[4] N. R. Werthamer, N. R. Werthamer, and Drougas, Risk and Reward. Springer, 2009.
[5] K. Jensen, "The Expected Value of an Advantage Blackjack player," Master of Science (MS), Utah State University, 2014.
[6] D. Rickwood, A. Blaszczynski, P. Delfabbro, N. Dowling, and K. Heading, "The psychology of gambling," InPsych, vol. 32, no. 6, pp. 11-21, 2010.
[7] A. W. Chau, J. G. Phillips, and K. L. Von Baggo, "Departures from sensible play in computer blackjack," The Journal of general psychology, vol. 127, no. 4, pp. 426-438, 2000. |
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